Hall Subgroups and π-Separable Groups

AbstractLetGbe a finite group and π a set of primes. In this paper, some new criteria for π-separable groups and π-solvable groups in terms of Hall subgroups are proved. The main results are the following:Theorem.G is aπ-separable group if and only if(1)G satisfies Eπand Eπ′;(2)G satisfies Eπ∪{q}and Eπ′∪{p}for all p∈π,q∈π′.Theorem.G is aπ-separable group if and only if(1)G satisfies Eπand Eπ′;(2)G satisfies Ep,qfor all p∈πand q∈π′